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User blog:Bubby3/Walkthrough of BMS.
Welcome to my walkthorough of BMS. On my analysis of BMS, I've got a LOT of critism for only providing a list of ordinals and not providing justification for my comparisons. So, on this blog post, I am going to give you a through analysis of BMS, with more reasons than actual comprarisons. I am going to start at trio sequence sysem, because as of now, it is universally accepted what the levels of primative and pair sequence system. So here is my analysis (of BM2.3). I am using Bucholoz's OCF (at least for this part) So, I am going to assume you know that (0,0,0)(1,1,1) has level \(\psi(\Omega_\omega)\). Here is some terminology. In the hydra representation, in (a,b,c), a corresponds to the height, and two entries are on top of each other if the height of the second one is greater than that of the first one, and there is no entry in between them with height less than or equal to the height of the first entry. My definition of BM2.3 is here (Look at the table and where BM2.3 is, and find the subrules, don't clock the link) Part 1: (0,0,0)(1,1,1) to (0,0,0)(1,1,1)(2,1,1) Adding more (1,1,1)'s You obviosly know that, from pair sequence system, that (x,1) or (x,1,0) corresponds \(\Omega\). So adding (1,1,0)'s adds \(\Omega\)'s to the OCF represtation. So, (0,0,0)(1,1,1)(1,1,0) has level (\psi(\Omega_\omega + \Omega)\). We can even stack (x,1,0)'s to get power towers of \(\Omega\)'s, or \(\psi_1\), until we get (1,1,0)(2,2,0), which has level \(\varepsilon_{\Omega+1}\). Becasue trio sequence works, this works until (1,1,0)(2,2,1), which has level \(\psi_1(\Omega_\omega)\) You might be asking, why didn't we add any \(\Omega_2\)'s until now. It's because in a (x,2,0) entry, x must be at least 2, so it must have at least 2. Also, for (x,2,0) to behave as \(\Omega_2\), it must be on top of, or a descendend of a (x,1,0) seperator, So, adding an \(\Omega_2\) to \(\psi(\Omega_\omega)\) is delayed until (0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0). Now (x,2,0) corresponds to an \(\Omega_2\). We can have things on top of (2,2,0) to have \(\psi_2\)\. This pattern repeats with (x,3,0)'s, (x,4,0)'s, until we get to (0,0,0)(1,1,1)(1,1,1), which corresponds to adding another \(\Omega_\omega\) to the psi function. This process repeats after (0,0,0)(1,1,1)(1,1,1) to get to (0,0,0)(1,1,1)(1,1,1)(1,1,1), so (1,1,0)(2,2,1)(2,2,1) corresponding to \(psi_1(\Omega_\omega 2)\), (2,2,0)(3,3,1)(3,3,1) corresponding to \(\psi_2(\Omega_\omega 2)\), etc. That means that (0,0,0)(1,1,1)(1,1,1)(1,1,1) corredponds to \(\psi(\Omega_\omega 3)\). So, adding (1,1,1)'s means adding \(\Omega_\omega\) s to the psi function. This process can even be prepeated transfinitely, until (0,0,0)(1,1,1)(2,1,0), because adding (1,1,1)'s does not mess with the second entry of (2,0,0)'s or it's desendents. Upgrade efffect The pattern of adding a \(\Omega_\omega\) when we add a (1,1,1) doesn't always hold after (0,0,0)(1,1,1)(2,1,0)(1,1,1). Sometimes, when the expression ends with a (x,1,0), it "upgrades" the \(\Omega\) to a \(\Omega_\omega\). The second entry of (x,1,0) gets increased by the (1,1,1), resulting in the (x,1,0) replaced by (x+1,2,0), which corresponds to \(\Omega_2\), then (x+3,3,0), which correponds to \(\Omega_3\), etc. The reason why this works is that (2,2,1) won't upgrade (x,1,0)'s. For example, in (0,0,0)(1,1,1)(2,1,0)(1,1,1), the second (1,1,1) upgrades the (2,1,0) when it is decomposing, resulting in (0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,2,0)(2,2,0)(3,3,1)(4,3,0). . However, in (0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1), the 1 in (3,1,0) does not increase when we decompose it, resuting in no infinite loop, so the level of (0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1) is \(\psi(\Omega_\omega \Omega + \Omega_\omega)\) So, this all means that (0,0,0)(1,1,1)(2,1,0)(1,1,1) has level \(\psi(\Omega_\omega^2)\) . Adding another (1,1,1) to that adds another \(\Omega_\omega\), because the upgrading still happens inside the (\psi_1)\ with the (2,2,1), in the form of (1,1,0)(2,2,1)(3,2,0)(2,2,1), so (1,1,0)(2,2,1)(3,2,0)(2,2,1) has level \(\psi_1(\Omega_\omega^2)\), which means we still add a \(\Omega_\omega\) Category:Blog posts